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M9302 Mathematical Models in Economics
Instructor: Georgi Burlakov
4.1.Dynamic Games of Incomplete Information
Lecture 6 05.05.2011
How to solve the GT problem?
Perfect Bayesian Equilibrium (PBNE)
in static games of complete information
in dynamic games of complete information
in static games of incomplete information
in dynamic games of incomplete information
Strategic Dominance Nash Equilibrium (NE)
Solution Concepts:
Backwards Induction Subgame-Perfect Nash Equilibrium (SPNE)
Bayesian Nash Equilibrium (BNE)
How to describe a dynamic game?
The extensive form representation of a game specifies:
1. Who are the PLAYERS.
2.1. When each player has the MOVE.
2.2. What each player KNOWS when she is on a move.
2.3. What ACTIONS each player can take.
3. What is the PAYOFF received by each player.
How to describe a game of incomplete information?
A game in which one of the players does not know for sure the payoff function of the other player is a game of INCOMPLETE INFORMATION
Thanks to Harsanyi (1967) games of incomplete information could be represented as dynamic games of complete but imperfect information
For the purpose, in the first stage a neutral player (Nature) is introduced to decide what will be the type (payoffs) of the players which is private information for at least one of them.
PLAYERS ACTIONS – A1, … ,An; Ai = {ai1,…, ain}
TYPES – Ti = {ti1,…, tin} System of BELIEFS ‑ PAYOFFS ‑ which is briefly denoted as
Normal-form Representation
)/( iii ttp
);,...,( 1 ini taau
},...,;,...,;,...,;,...,{ 1111 nnnn uuppTTAAG
The aim of the sixth lecture is to show:
1. How to strengthen the Bayesian equilibrium to hold in dynamic games?
2. How to define the resulting solution concept?
The static (simultaneous-move) games
Example: Students’ Dilemma - 4
Strategic behaviour of students taking a course:
First, YOU and OTHERS might be called on to choose between studying HARD or taking it EASY but YOU could reject if YOU feel UNSURE. Then, the game ends.
If YOU do not reject, students do the exam and get a grade.
Example: Students’ Dilemma
Standard assumptions: Students choose between HARD and EASY
SIMULTANEOUSLY. Grading is announced in advance, so it is
COMMON KNOWLEDGE to all the students.
Simplification assumptions: Performance depends on CHOICE.
EQUAL EFFICIENCY of studies.
Example: Students’ Dilemma
Normal Form Representation:
1. 2 players – YOU vs. OTHERS2. Single choice strategies
3. Payoff function:
},{
},,{
HardEasyS
UnsureHardEasyS
OTHERS
YOU
{-1.5} PayoffRejection
,,
i
iiiiiiiii
ssGRADEsLEISUREssuu
Example: Students’ Dilemma
Grading Policy:
the students over the average have a STRONG PASS (Grade A, or 1),
the ones with average performance get a WEAK PASS (Grade C, or 3) and
who is under the averageFAIL (Grade F, or 5).
Example 1: Students’ Dilemma – 4
Leisure Rule: HARD study schedule devotes all the time (leisure = 0) to studying distinct from the EASY one (leisure = 1).
Player i’s choice
Others’ choice LEISURE GRADE Player i’ payoff
Easy All Easy 1 -2 -1
At least one Hard 1 -5 -4
Hard At least one Easy 0 -1 -1
All Hard 0 -2 -2
Rejection -1.5
Example 1: Students’ Dilemma – 3
Game Tree VS. Normal-FormYOU
OTHERS OTHERS
-1-1
-4-1
-2-2
EASY HARD
EASY -1,-1(NE)
-4,-1
HARD -1,-4 -2,-2
UNSURE -1.5,-1.5 -1.5,-1.5 (NE)
-1-4
Easy Hard
Easy Hard Easy Hard
Unsure -1.5,-1.5
Example 1: Students’ Dilemma – 4
There is a single subgame with 2 NE: {E, E} and {U, H}
However, if YOU do not reject,{E,E} is not the only equilibrium
How to strengthen the solution concept to allow for strict prediction?
Dynamic Games of Incomplete Information
One way to strengthen the equilibrium concept is to impose the following two requirements:Requirement 1: At each information set the player with the move must have a belief about which node in the information set has been reached by the play of the game.Requirement 2: Given their beliefs, the players’ strategies must be sequentially rational. That is, at each information set the action taken by the player with the move must be optimal given the player’s belief and the other player’s subsequent strategies.
Example 1: Students’ Dilemma – 4
Given OTHER’s belief that YOU would move EASY with probability p, their expected payoff of playing: EASY is:
HARD is:
HARD weakly dominates EASY which rules out {E,E} for p<1.
43141)( pppEASYE OTHERS
2121)( pppHARDE OTHERS
Dynamic Games of Incomplete Information
Requirements 1 and 2 insist that the players have beliefs and act optimally given these beliefs, but not that these beliefs be reasonable. For the solution to be strict, the following requirement must also hold:Requirement 3: At information sets on the equilibrium path, beliefs are determined by Bayes’ rule and the players’ equilibrium strategies.
Definition: For a given equilibrium in a given extensive-form game, an information set is on (off) the equilibrium path if it will (not) be reached with positive probability when the game is played according to the equilibrium strategies.
Example 1: Students’ Dilemma – 4
If the Nash equilibrium {U, H} holds, the belief of OTHERS should be p<1.
It is not reasonable for OTHERS to choose H if they believe that YOU will play E for sure (p=1).
Suppose there is a mixed-strategy equilibrium in which YOU plays E with probability q1, H with probability q2 and U with probability 1- q1- q2.
In conformity with the Bayes’ rule,OTHERS’ belief is forced by requirement 3 to be p = q1/(q1+q2).
Perfect Bayesian Equilibrium In simple dynamic games of incomplete
information – including the signaling game – requirements 1 through 3 constitute the definition of a perfect Bayesian equilibrium.
In richer games, however, more requirements need to be imposed: Requirement 4: At information sets off the
equilibrium path, beliefs are determined by Bayes’ rule and the players’ equilibrium strategies.
Definition: A perfect Bayesian equilibrium consists of strategies and beliefs satisfying Requirements 1 through 4.
Summary In order to rule out non-credible threats or promises,
the Bayesian equilibrium concept needs to be strengthened.
In the definition of the perfect Bayesian Nash equilibrium (PBNE) beliefs are elevated to the level of importance of strategies.
For the simple games of incomplete information, PBNE consists of 3 basic requirements on players’ beliefs.
Richer dynamic games have more specific requirements concerning the beliefs off the equilibrium path that also need to be satisfied.